The problem of the last model was the combination of coloring and revisiting
steps. Those must be separated. Therefore V. Lonati and M. Pradella presented
another model~\cite{lonati2011towards}. This model separats the two phases that
are called tiling and roaming. In the tiling phase the automaton simulates a
$\mu$-DWA and in the roaming phase the behaviour of a 4DFA is simulated. That
means the automaton executes a sequence of coloring steps at first and then it
performs a sequence of revisiting steps, using the information enclosed in the
colors placed before. The second phase starts at the position where the first
phase has ended. This model is called 4-way deterministic $\mu$-directed Wang
automaton and is abbreviated as $\mu$-4DWA.
\begin{definition}
A \emph{4-way deterministic $\mu$-directed Wang automaton} is a 9-tuple \\
$M = (Q, \Sigma, C, \gamma, \mu, \delta, q_0, q_e, q_r)$, where
\begin{compactitem}
  	\item $Q$ is a finite set of states, 
	\item $\Sigma$ is a finite input alphabet,
	\item $C$ is a finite set of colors,
	\item $\gamma: \Sigma_C \times Dirs \rightarrow \Sigma_{4C}$, is the tiling
	transition function, is a partial function such that each tile in $\gamma(A,
	d)$ extends A,
	\item $\mu = (\eta, c_s, d_s)$ is a polite scanning strategy such that for all
	tiles $A$ and directions $d$ with $\gamma(A, d)\neq \emptyset$ implies
	$\eta(\Delta_A, d) \neq \perp$,
	\item $\delta: \Sigma_{4C} \times Q \rightarrow Q \times Dirs$ is
	the roaming transition function and
	\item $q_0, q_e, q_r$ are the initial roaming, accepting and rejecting states.
\end{compactitem}
\end{definition}
\begin{example}
A $\mathcal{S}^{t2b}$-4DWA $M$ can accept the language $L_{fr = r'} \cap
L_{half}$. $L_{fr = r'}$ is the language defined in Section~\ref{urec}.
$L_{half}$ is the language described in Section~\ref{scan_drec}. That $L(M) =
L_{fr = r'} \cap L_{half}$ follows, $M$ must check the input picture by
simulating the $\mathcal{S}^{t2b}$-DWA from
Example~\ref{example_mu_directed_wang_automaton} to verify that the input
picture is in $L_{fr = r'}$ and then simulates the 4DFA
from~\cite{ito1988threeway} to check if the input picture is in $L_{half}$.
\end{example}
\begin{theorem}
For every polite $\mu$, $\familyOf{$\mu$-4DWA}$ is a boolean algebra.
\end{theorem}
This is given by definition and that $\familyOf{4DFA}$ and
$\familyOf{$\mu$-DWA}$ are boolean algebras~\cite{lonati2010deterministic}. 

The following theorem is also proved by the introducers of this automaton model
in~\cite{lonati2011towards}. 
\begin{theorem}
For every polite $\mu$, $\familyOf{4DFA} \cup
\familyOf{$\mu$-DWA} \subset \familyOf{$\mu$-4DWA} \subseteq REC$
\end{theorem}
In the proof, they wrote that 4DFA and $\mu$-DWA are special kinds of $\mu$-4DWA
which implies the first relationship. The second relationship is proved by a
construction an abitrary language $L$ that is accepted by some $\mu$-4DWA in
such a way that $L \in$ REC.